Multi-hop visibility through the vacant set of Poissonian obstacles

Abstract

We study multi-hop visibility inside the vacant set of three obstacle models in ${\mathbb{R}}^d$ with slow decay of spatial correlations and disparate obstacle geometries: Poisson-Boolean models with general i.i.d. radii distributions, Poisson cylinders and Brownian interlacements. For any $N\ge 0$, we obtain sharp bounds on the probability $P_{\mathrm{vis}}^N(r)$ of visibility to distance $r$ via $(N+1)$ hops in terms of the (explicit) probability of direct visibility to distance $r$ in a given direction, generalizing our earlier result from arXiv:2304.10298 for $N=0$. We observe a universal behavior of $P_{\mathrm{vis}}^N(r)$ in terms of two characteristic scales. In the three models of interest, these scales are generally the same, but with some anomalous exceptions in low dimensions.

36 pages